# Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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### First homology group of the general linear group

The abelianization of the general linear group $GL(n,\mathbb{R})$ defined by $GL(n)^{ab} := GL(n)/[GL(n), GL(n)]$ is isomorphic to $\mathbb{R}^{\times}$. This follows from the fact that $[GL(n),GL(n)] ...

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### Topological derivation of the Laplace formula for determinants in Euclidean space

I'm interested in trying to derive the Leibniz formula for the determinant on real Euclidean space, without first constructing $\det$ by axiomatizing its properties.
We know $GL(n)$ deformation ...

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### Reference Request: Carnot Group Not Containing Group of Isometries

This question is a follow-up to this post, from which I quote:
Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is ...

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### Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$

Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...

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### Interpretation of determinants on commutative rings

In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map.
This interpretation conceptually depends ...

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### History of the notion of $(G,X)$-structure

I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...

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### Lattices are not solvable in non-compact semisimple Lie groups

I'm trying to prove the following result.
If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...

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### Computing Deligne-Lusztig Characters in General

The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the ...

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### Generalizing polynomial identities for rings

For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...

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### Independence of Duistermaat-Heckman measure

Suppose that a compact K?hler manifold $(X,\omega)$ has a real torus acting on it by symplectomorphisms in a Hamiltonian way (the torus is not necessarily of maximal rank). Then for any smooth ...

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### Reference Request: Carnot Groups over Complexes

Is there a theory of complex (analytic) Carnot groups and Caratheodory metrics?

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### On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not

In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...

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### Nonnegativity of an integral over the unitary group

For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...

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### Uniqueness of Equivariant Harmonic Map for Surface Group Representation

In section 1.2 of https://arxiv.org/pdf/1311.2919.pdf the following result is stated.
$\textbf{Theorem}$ (Labourie). Let $S$ be a closed Riemann surface of negative Euler characteristic, $Γ$ its ...

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### On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya

I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...